An adjoint method to calculate the gradient of island width in stellarators is presented and applied to a set of magnetic field configurations. The underlying method for calculation of the island width is that of Cary & Hanson (Phys. Fluids B, vol. 3, issue 4, 1991, pp. 1006–1014) (with a minor modification), and requires that the residue of the island centre be small. Therefore, the gradient of the residue is calculated in addition. Both the island width and the gradient calculations are verified using an analytical magnetic field configuration introduced by Reiman & Greenside (Comput. Phys. Commun., vol. 43, issue 1, 1986, pp. 157–167). The method is also applied to the calculation of the shape gradient of the width of a magnetic island in a National Compact Stellarator Experiment (NCSX) vacuum configuration with respect to positions on a coil. A gradient-based optimization is applied to a magnetic field configuration studied by Hanson & Cary (Phys. Fluids, vol. 27, issue 4, 1984, pp. 767–769) to minimize stochasticity by adding perturbations to a pair of helical coils. Although only vacuum magnetic fields and an analytical magnetic field model are considered in this work, the adjoint calculation of the island width gradient could also be applied to a magnetohydrodynamic (MHD) equilibrium if the derivative of the magnetic field, with respect to the equilibrium parameters, is known. Using the island width gradient calculation presented here, more general gradient-based optimization methods can be applied to design stellarators with small magnetic islands. Moreover, the sensitivity of the island size may itself be optimized to ensure that coil tolerances, with respect to island size, are kept as high as possible.

中文翻译：

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一种确定岛尺寸对磁场变化敏感性的辅助方法

提出了一种计算仿星器中岛宽度梯度的伴随方法，并将其应用于一组磁场配置。计算岛宽度的基本方法是 Cary & Hanson (物理。流体B，卷。3, issue 4, 1991, pp. 1006–1014)（稍作修改），并要求岛中心的残余物很小。因此，还计算了残差的梯度。岛宽度和梯度计算均使用 Reiman & Greenside 引入的分析磁场配置进行验证（计算。物理。交流。, 卷。43，1986 年第 1 期，第 157-167 页）。该方法还应用于计算国家紧凑型仿星实验 (NCSX) 真空配置中磁岛宽度相对于线圈位置的形状梯度。基于梯度的优化应用于 Hanson & Cary 研究的磁场配置 (物理。流体, 卷。27, issue 4, 1984, pp. 767–769) 通过向一对螺旋线圈添加扰动来最小化随机性。虽然在这项工作中只考虑了真空磁场和解析磁场模型，但如果磁场的导数相对于平衡参数，岛宽度梯度的伴随计算也可以应用于磁流体动力学 (MHD) 平衡, 是已知的。使用此处介绍的岛宽度梯度计算，可以应用更通用的基于梯度的优化方法来设计具有小磁岛的仿星器。此外，岛尺寸的敏感性本身可以被优化以确保线圈公差相对于岛尺寸保持尽可能高。